THINK: Go back to the star-like image and manipulate it to get a very high r-ratio. What does the figure (image) look like? Try to get the ratio as high as you possibly can without crossing the lines. (Java File)

All squares have r = 16. | Java File

All circles have r = 4*pi = 12.56... | Java File

All 4x5 (8x10) rectangles have r = 16. 2. | Java File

THINK: Triangles are figures that are unique in this ratio and are relatively easy to explore. The sketch on the next page offers the ability to explore a triangle between two parallel lines. Carefully note the area of the triangle as you move the top vertex, what happens to the perimeter? What about the ratio (r)?

After exploring general triangles we can explore "special" triangles:

All 3-4-5 (6-8-10) rt. triangles have r = 24. | Java File

All equilateral triangles have r = 12*sqrt(3) = 20.78 | Java File

All 45-45-90 rt triangles have r = 12 + 8sqrt(2) = 23.3. | Java File

Conclusion: We have explored the cancer ratio (r) with figures up to four sides. Establish the ratio for sides with more than four and try to create a figure with a large ratio and small ratio. What types of figures produce the largest ratio? What types of figures produce the smallest ratios?

Think: Try various ratios out with this figure to experiment with the topics you just learned?